Radicals. 45 



23. Problem. To Rationalize any Binomial Quadratic 

 Surd. Any binomial quadratic surd may be represented by 

 a^p-^b^q, where a and b may be either positive or negative. 

 The rationahzing factor is plainly ay/])—b\^q, for 



(a-y'~p^bs/~q){a^p—b\^~q)^a'p—b'q, 

 which is rational. 



24. Problem. To Rationalize any Trinomial Quad- 

 ratic Surd. Any trinomial quadratic surd may be represented 

 by a>yp-\-b^q-\-c's^r, where a, b, and c are supposed to be any 

 rational quantities whatever, positive or negative or integral or 

 fractional. Multiply first by a^p-\-b\^ q—c^r, and we obtain 



(as/p-\-bs/q-\-c^'r){aVp-\-b\^q—c\^'r) = {aVp-irb^~qY—{c\^rY 

 -=a'p-\-b'q—c^r-\-2ab^pq, (i) 



which is rational as far as r is concerned. Now multiply this by 



(a-p-^b-q—c^r) — 2ab^pq (2) 



and we obtain 



{(a'p^lfq—c'r)-\-2abypq]{{a''p-\-b'q—c^r) — 2abypq] 



= (a^p + b^q-c^r(r-_^a^b^pq, (3) 



which is rational with respect to all the quantities. The ration- 

 alizing factor for the original trinomial quadratic surd is thus 

 seen to be 



ia yp-\-byq—c\^r){a'p-hb^-q—c'r—2abs^p^)^ (4) 



25. The second parenthesis in (4) above will be found to be 

 composed of the two factors 



(a\/])—b\^q-\-c\^r)(a\/])—b^q—c\^r) 

 Hence the rationalizing factor of a\^ p -\-by/ji^^^C\/fr j^ay be 

 written 

 (a>/p -\-b\^q —c^r)(as^p—b\^q -\-)s(^yr^)(a>/p—b>/q —c>/r) 

 Observe that the terms of each of the component trinomial fac- 

 tors of this expression are those of the given irrational quantity 

 and the signs are those exhibited in the scheme — 



+ + - 

 + - + 



if 



